Is zakat obligatory on a sum of money, given that its owner is about to get married, does not own a house or furniture, and his father has passed away leaving a large debt, a significant portion of which he may have to bear?

Zakat is obligatory on wealth that has reached the nisab (minimum threshold) and on which a full lunar year has passed, even if it has been set aside for marriage or Hajj, because it is a right of the poor and purifies wealth. As for the portion of your father's debt that you are bearing, it is deducted from your wealth if the debt is due immediately or deferred for a non-distant period. If, after deduction, the wealth falls below the nisab, then no Zakat is due on it; otherwise, Zakat becomes obligatory.